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K. A. Saaifan, Jacobs University, Bremen
3. Voltage and Current laws
3.1 Node, Branches, and loops
A branch represents a single element such as a voltage source or a
resistor
A node is the point of the connection between two or more
elements (branches)
It is usually indicated by a dot in a circuit
If a connecting wire (short circuit) connects two nodes, the two nodes
constitute a single nodes
A loop is any closed path in a circuit
A closed path is formed by starting at a node, passing through a set of
nodes and returning to the start node without passing through any
node more than once
branch
loop
K. A. Saaifan, Jacobs University, Bremen
3.2 Kirchhoff's Current Law
Kirchhoff's Current Law (KCL) is based on the law of conservation
of charge
The algebraic sum of the currents entering any node is zero
An alternative form of KCL is “the current entering any node =
the current leaving that node”
KCL can be applied to any closed boundary (closed path)
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K. A. Saaifan, Jacobs University, Bremen
3.3 Kirchhoff's Voltage Law
Kirchhoff's voltage Law (KVL) is based on the law of conservation of
energy
The algebraic sum of the voltages around any closed path is zero
KVL
When voltage sources are connected in series, KVL can be applied to
obtain the total voltage
a
a
=
b
b
K. A. Saaifan, Jacobs University, Bremen
Determine vx in the circuit
Ans:
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K. A. Saaifan, Jacobs University, Bremen
Determine vx in the circuit
Ans:
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K. A. Saaifan, Jacobs University, Bremen
3.4 The Single-Loop Circuit
Single-loop circuits
Elements are connected in series
All elements carry the same current
We shall determine
The current through each element
The voltage across each element
The power absorbed by each element
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K. A. Saaifan, Jacobs University, Bremen
3.4 The Single-Loop Circuit
We apply the following steps
1) Assign a reference direction for the unknown current
2) Assign voltage references to the elements
3) Apply KVL to the closed loop path
KVL
4) Use Ohm's law where needed to get an equation in “i”
5) Solve for i
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K. A. Saaifan, Jacobs University, Bremen
Find i and p for all elements in the circuit
Ans:
KVL
1) Assign a reference direction for the unknown current
2) Assign voltage references to the elements (note that vA=-v2)
3) Apply KVL to the closed loop path
4) Use Ohms law where needed to get an equation in “i”
5) Solve for i
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K. A. Saaifan, Jacobs University, Bremen
Find i and p for all elements in the circuit
Ans:
KVL
Computing the power absorbed by each element
The total power absorbed by all elements
K. A. Saaifan, Jacobs University, Bremen
3.5 The Single-Node-Pair Circuit
Single-node-pair circuits
Elements are connected in parallel
All elements have a common voltage
We shall determine
The current through each element
The voltage across each element
The power absorbed by each element
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K. A. Saaifan, Jacobs University, Bremen
3.5 The Single-Node-Pair Circuit
We apply the following steps
1) Define the voltage v and arbitrary select its polarity
2) Use passive sign convention to determine the currents directions
3) Apply KCL at the node
4) Use Ohm's law where needed to get an equation in “v”
5) Solve for v
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K. A. Saaifan, Jacobs University, Bremen
Find v and p supplied by the independent source
Ans:
1) Assign an arbitrary sign for the unknown voltage
2) passive sign convention to find the currents directions (note that ix=-i2)
3) Apply KCL to the nodes
4) Use Ohm's law where needed to get an equation in “v”
5) Solve for v
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K. A. Saaifan, Jacobs University, Bremen
HW: Find i1, i2, i3, and i4
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K. A. Saaifan, Jacobs University, Bremen
3.6 Series and Parallel Connected Sources
Series-connected voltage sources can be replaced by a single
source
Parallel current sources can be replaced by a single source
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K. A. Saaifan, Jacobs University, Bremen
3.7 Resistors Series and Parallel
Series connection
KVL
Parallel connection
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K. A. Saaifan, Jacobs University, Bremen
Find the voltage and the power of the independent source
1) Apply KCL at the top node
2) Use Ohm's law for (i1=vx/6) and (vx=3i3)
3) Solve i3 and vx
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K. A. Saaifan, Jacobs University, Bremen
3.8 Voltage and Current Division
Voltage divider: is a passive linear circuit that produces an output voltage
(vout) that is a fraction of its input voltage (vin)
Easily solved with KCL, KVL, & equivalent
resistances
Then,
Generally, assume we have
The voltage vN can be given as
Easy to find the other voltages, too
K. A. Saaifan, Jacobs University, Bremen
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K. A. Saaifan, Jacobs University, Bremen
Current divider: is a simple linear circuit that produces an output current (iout)
that is a fraction of its input current (iin)
Easily solved with
Since
For n=2, we have
The circuit divider reduces to
K. A. Saaifan, Jacobs University, Bremen
Use resistance combination methods and current division
to find i1 and i2 and vx
Ans:
We note i1 goes to the following equivalent resistor
Use current divider, we have
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K. A. Saaifan, Jacobs University, Bremen
We note i2 goes to the following equivalent resistor
Use current divider, we have
HW: Solve vx
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K. A. Saaifan, Jacobs University, Bremen
We note i2 goes to the following equivalent resistor
Use current divider, we have
HW: Solve vx
Homework Assignment 2
P3.6, P3.7, P3.13, P3.15, P3.16, P3.19, P3.20, P3.21, P3.30, P3.31, P3.35,
P3.39, P3.73, P3.75 and P3.82
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